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DoF-Robust Strategies for the K-user
Distributed Broadcast Channel with Weak CSI
Paul de Kerret and David Gesbert
Communication Systems Department, EURECOM, Sophia-Antipolis, France
Abstract
In this paper1 we consider the Network MIMO channel under the so-called Distributed Channel
State Information at the Transmitters (D-CSIT) configuration. In this setting, the precoder is designed
in a distributed manner at each Transmitter (TX) on the basis of the locally available multi-user channel
estimate. Although the use of simple Zero-Forcing (ZF) was recently shown to reach the optimal DoF
for a Broadcast Channel (BC) under noisy, yet centralized, CSIT, it can turn very inefficient in the
distributed setting as the achieved number of Degrees-of-Freedom (DoF) is then limited by the worst
CSI accuracy across all TXs. To circumvent this effect, we develop a new robust transmission scheme
improving the DoF. A surprising result is uncovered by which, in the regime of so-called weak CSIT,
the proposed scheme is shown to achieve the centralized outerbound obtained under a genie-aided
centralized setting in which the CSI versions available at all TXs are shared among them. Building
upon the insight obtained in the weak CSIT regime, we develop a general D-CSI robust scheme which
improves over the DoF obtained by conventional ZF approach in an arbitrary CSI quality regime and
is shown to achieve the centralized outerbound in some other practically relevant CSIT configurations.
I. INTRODUCTION
Multiple-antennas at the TX can be exploited to serve multiple users at the same time, thus
offering a strong DoF improvement over time-division schemes [1]. This DoF improvement is
however critically dependent on the accuracy of the CSIT. Indeed, the absence of CSIT is known
1D. Gesbert and P. de Kerret are supported by the European Research Council under the European Union’s Horizon 2020
research and innovation program (Agreement no. 670896).
This paper has been presented in part at the 2016 Information Theory and Applications Workshop and at the 2016 IEEE
International Symposium on Information Theory.
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to lead to the complete loss of the DoF improvement in the case of a BC with symmetric users [2].
In the noisy (centralized) CSIT regime, a long standing conjecture by Lapidoth, Shamai, and
Wigger [3] has been recently settled in [4] by showing that a scaling of the CSIT error in P−α
for α ∈ [0, 1] leads to a DoF of 1 + (K − 1)α in the K-user BC.
A different line of work in the area of BC with limited feedback has been focused on the
exploitation of delayed CSI on the TX side. This research area was triggered by the seminal
work from Maddah-Ali and Tse [5] where it was shown that completely outdated CSIT could
still be exploited via a multi-phase protocol involving the retransmission of the interference
generated. While the original model considered completely outdated CSIT, a large number of
works have developed generalized schemes for the case of partially outdated [6], [7], alternating
[8], or evolving CSIT [9], to name just a few.
In all the above literature, however, centralized CSIT is typically assumed, i.e., precoding
is done on the basis of a single imperfect/outdated channel estimate being common at every
transmit antenna. Although meaningful in the case of a BC with a single transmit device, this
assumption can be challenged when the joint precoding is carried out across distant TXs linked
by heterogeneous and imperfect backhaul links, as in the Network MIMO context. Such a setting
could in particular be obtained if the user’s data symbols are cached at the different TXs before
the transmission [10]. In this case, it is expected that the CSI exchange will introduce further
delay and quantization noise such that it becomes necessary to study the impact of TX dependent
CSI noise.
In order to account for TX dependent limited feedback, a distributed CSIT model (here
referred to as D-CSIT) was introduced in [11]. In this model, TX j receives its own multi-
user imperfect estimate H(j) on the basis of which it designs its transmit coefficients, without
additional communications with the other TXs. The finite-SNR performance of regularized ZF
under D-CSIT has been computed in the large system limit in [12] while heuristic robust
precoding schemes have been provided in [13], [14] for practical cellular networks. In [15],
[16], Interference Alignment is studied in a D-CSIT configuration and methods to reduce the
required CSIT gaining at each TX are provided. The DoF achieved with delayed and local CSIT,
which is hence a particular D-CSIT configuration is also studied in several works [17], [18], but
the results provided are restricted to the particular local CSIT configuration considered.
In terms of DoF, it was shown in a previous work [11] that using a conventional ZF precoder
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(regularized or not) leads to a severe DoF degradation caused by the lack of a consistent CSI
shared by the cooperating TXs. A scheme was proposed to lift the DoF in the two-user case but
the scheme was relying on the particular structure of the 2-user case and could not be extended
to more users. In this paper, we further build up on this concept to establish a general strategy
for robust precoding in the distributed setting.
More precisely, the main findings read as follows.
• We show that optimal precoding strategies differ depending on the level of CSI accuracy
available at the TXs. To this end we differentiate a weak CSIT regime where the accuracy
at the least informed TX lies below a specific threshold, from an arbitrary CSIT regime
where such condition does not hold.
• In the weak CSIT regime –which will be defined rigorously below– we obtain the surprising
result that it is possible to reach the centralized outerbound, where the centralized outerbound
is obtained by centralizing all CSI feedback observations across all TXs.
• In the arbitrary CSIT regime, the above D-CSIT robust scheme is extended, helping lift the
DoF substantially above what is achieved by conventional ZF precoding and achieving the
centralized outerbound in several key CSIT configurations.
Notations: We denote the multivariate circularly symmetric complex Gaussian distribution
with zero mean and identity covariance matrix by NC(0, I). We use .= to denote exponential
equality, i.e., we write f(P ).= P x to denote limP→∞
log f(P )logP
= x. The exponential inequalities
≤ and ≥ are defined in the same way. We also use the shorthand notation [K] to denote the set
{1, . . . , K}.
II. SYSTEM MODEL
A. Transmission Model
We study a communication system where K TXs jointly serve K Receivers (RXs) over a
Network (Broadcast) MIMO channel. We consider that each TX is equipped with a single-
antenna. Each RX is also equipped with a single antenna and we further assume that the RXs
have perfect CSI so as to focus on the impact of the imperfect CSI on the TX side.
The signal received at RX i is written as
yi = hHi x + zi (1)
4
where hHi ∈ C1×K is the channel to user i, x ∈ CK×1 is the transmitted multi-user signal, and
zi ∈ C is the additive noise at RX i, being independent of the channel and the transmitted signal,
and distributed as NC(0, 1). We further define the channel matrix H , [h1, . . . ,hK ]H ∈ CK×K
and the channel coefficient from TX j to RX i as Hi,j . The channel is assumed to be drawn from
a continuous ergodic distribution such that all the channel matrices and all their sub-matrices
are full rank with probability one.
The transmitted multi-user signal x is obtained from the symbol vector s ∈ Cb×1 having its
elements Independently and Identically Distributed (i.i.d.) according to NC(0, 1), where b is the
number of independent data symbols emitted, through joint precoding.
B. Distributed CSIT Model
The D-CSIT setting differs from the conventional centralized one in that each TX receives a
possibly different (global) CSI based on which it designs its own transmission parameters without
any additional communication to the other TXs. Specifically, TX j receives the imperfect multi-
user channel estimate H(j) = [h(j)1 , . . . , h
(j)K ]H ∈ CK×K where (h
(j)i )H refers to the estimate of
the channel from all TXs to user i, at TX j. TX j then designs its transmit coefficients solely
as a function of H(j) and the statistics of the channel.
Remark 1. It is critical to this work to understand well how the distributed CSIT setting differs
from (embeds) the many different heterogeneous CSIT configurations studied in the literature.
Indeed, an heterogeneous CSIT configuration typically refers to a centralized CSIT configuration
(i.e., with a common channel estimate at all TXs), where each element of the channel is known
with a different quality owing to specific feedback mechanisms. In contrast, the distributed
setting considered here has as many different channel estimates as there are TXs (where each
TX does not have access to the CSIT knowledge at the other TXs), and where different channel
coefficients may also be represented with unequal quality (as in heterogeneous case).
We model the CSI uncertainty at the TXs as
H(j) = H +√P−α(j)∆(j) (2)
where ∆(j) is a random variable with zero mean and bounded covariance matrix. The scalar
α(j) is called the CSIT scaling coefficient at TX j.
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Remark 2. The CSIT scaling coefficient α(j) takes its value in [0, 1] where α(j) = 0 is generally
seen to correspond to a CSIT being useless in terms of DoF. In contrast, α(j) = 1 is usually
equivalent in terms of DoF to a perfect CSIT [4], [19].
The multi-user distributed CSIT configuration is represented through the multi-user CSIT
scaling vector α ∈ RK defined as
α ,
α(1)
...
α(K)
. (3)
For ease of notation, we also define the maximum value of these CSIT scaling coefficients
αmax , maxj∈[K]
α(j). (4)
In addition, we consider that the channel realizations and the channel estimates are drawn in an
i.i.d manner. For a given transmission power P , we further assume that the conditional probability
density functions also verify that
maxH∈CK×K
(pH|H(1),...,H(K)(H)
).=√Pαmax . (5)
Remark 3. This condition extends to the distributed CSIT configuration the condition provided
in [4], which writes in our setting as
maxH∈CK×K
(pH|H(j)(H)
).=√Pα(j) , ∀j ∈ [K]. (6)
Condition (5) is a mild technical assumption, which holds for the distributions usually con-
sidered.
Example 1. We show in the Appendix that condition (5) is satisfied when the noise realiza-
tions ∆(j) ∈ CK×K are i.i.d. according to NC (0K , IK) and all the CSI noise error terms ∆(j)
are independent of each other. �
This D-CSIT setting is illustrated in Fig. 1.
6
K TXs
K users
KK x )( Cˆ jH
K x 1H Cih
)( j
s
K Cx
KK x CH
Fig. 1: Network MIMO with Distributed CSIT
C. Degrees-of-Freedom Analysis
Let us denote by C(P ) the sum capacity of the D-CSIT network MIMO channel above. The
optimal sum DoF in this distributed CSIT scenario is denoted by DoFDCSI(α) and defined by
DoFDCSI(α) , limP→∞
C(P )
log2(P ). (7)
III. A TOY EXAMPLE
We start by presenting a simple transmission scheme in a Toy-example as it contains some
important features of the setting and allows to convey the main intuition in a clear manner. This
motivating scheme will then be improved to obtain the results in Section IV.
Let us then consider a 3-user setting in which α(1) = 0.1, α(2) = 0, and α(3) = 0. We will
show how it is possible to achieve the DoF of 1 + 2α(1) = 1.2, which is the value of the DoF
that would be achieved in a centralized setting with TX 2 and TX 3 having received the same
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estimate as TX 1 [19]. In fact, it will be rigorously shown in Section IV that the DoF obtained
when forming such a centralized setting is always an outerbound.
A. Encoding
The transmission scheme consists in a single channel use during which 3 private data symbols
of rate α(1) log2(P ) bits are sent to each user (thus leading to 9 data symbols being sent in one
channel use), while an additional common data symbol of rate (1−α(1)) log2(P ) bits is broadcast
from TX 1 to all users using superposition coding [20]. Note that the information contained in
this common data symbol is not only composed of “fresh” information bits destined to one user,
but is also composed of side information necessary for the decoding of the private data symbols,
as will be detailed below.
The transmitted signal x ∈ C3 is then equal to
x = s1 + s2 + s3 +
1
0
0
s0 (8)
where
• si ∈ C3 is a vector containing private data symbols destined to user i, with power Pα(1)/3
and rate α(1) log2(P ) bits.
• s0 is the common data symbol transmitted by TX 1 only and destined to all users, with
power P − Pα(1) and rate (1− α(1)) log2(P ) bits.
The signal received at user i is then equal to
yi=Hi,1s0︸ ︷︷ ︸.=P
+ hHi s1︸ ︷︷ ︸.=Pα
(1)
+ hHi s2︸ ︷︷ ︸.=Pα
(1)
+ hHi s3︸ ︷︷ ︸.=Pα
(1)
(9)
where we have written under the bracket the power scaling, and where the noise term has been
neglected for clarity. The received signals at the users during this transmission are illustrated in
Fig. 2.
B. Interference Estimation and Quantization at TX 1
The key element of the scheme is that the common data symbol s0 is used to convey side
information allowing each user to decode their desired data symbols. More specifically, TX 1
8
Pα(1)
P
3H12
H11
H101,1 shshsh sH 3
H22
H21
H201,2 shshsh sH 3
H32
H31
H301,3 shshsh sH
P1-α(1)
RX 3RX 2RX 1
Fig. 2: Illustration of the received signals at the users.
uses its local CSIT H(1) to estimate the interference terms (h(1)i )Hsk∀i, k, k 6= i, quantize them,
and then transmit them using the common data symbol s0. Each interference term is quantized
using α(1) log2(P ) bits such that the quantization noise remains at the noise floor. Hence, the
transmission of all the quantized estimated interference terms requires to transmit 6α(1) log2(P )
bits.
These 6α(1) log2(P ) bits can be transmitted via the data symbol s0 if 6α(1) log2(P ) ≤ (1 −
α(1)) log2(P ), which is the case for the example considered here. It the inequality is strict,
the data symbol s0 transmits some additional (1 − 7α(1)) log2(P ) fresh information bits to any
particular user.
C. Decoding and DoF Analysis
It remains to verify that this scheme leads to the claimed DoF. Let us consider without loss of
generality the decoding at user 1 as the decoding at the other users will follow with a circular
permutation of the user’s indices. Note that signals at the noise floor will be systematically
omitted.
Using successive decoding [20], the data symbol s0 is decoded first, followed by the data
symbol s1. The data symbol s0 of rate of (1−α(1)) log2(P ) bits can be decoded with a vanishing
probability of error as its SINR can be seen in (9) to scale in P 1−α(1) .
Upon decoding s0, the estimated interferences (h(1)1 )Hs2 are obtained (up to the quantization
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noise at the noise floor, which is thus omitted). It remains to evaluate the impact over the DoF
of the imperfect estimation at TX 1:
(h(1)1 )Hs2 = hH
1 s2 +√P−α(1)(δ
(1)1 )Hsk︸ ︷︷ ︸
.=P 0
. (10)
It follows from (10) that the interference terms can be suppressed up to the noise floor at RX 1
using the quantized estimated interference terms received.
After having subtracted the quantized interference terms, the remaining signal at user 1 is then
y1 = hH1 s1. (11)
Using this signal in combination with the estimated interference terms (h(1)2 )Hs1 and (h
(1)3 )Hs1
obtained through s0, user 1 forms a virtual received vector yv1 ∈ C3 defined as
yv1 ,
hH
1
(h(1)2 )H
(h(1)3 )H
s1. (12)
Each component of yv1 has a SINR scaling in Pα(1) such that user 1 can decode with a vanishing
error probability its destined 3 data symbols having each the rate α(1) log2(P ) bits.
Considering the 3 users, it is possible to transmit in one channel use 9α(1) log2(P ) bits and
(1− 7α(1)) log2(P ) bits (through the data symbol s0), which yields a sum DoF of 1 + 2α(1).
It can be easily seen that this scheme is able to achieve the optimal DoF 1 + 2α(1) as long as
α(1) ≤ 1/7. In Section IV, this scheme will be improved by introducing some ZF precoding to
reduce the amount of interference to retransmit, thus allowing for larger values of α(1).
Remark 4. Interestingly, the above scheme builds on the principle of interference estimation,
quantization and retransmission, which has already been exploited in the different context of
precoding with delayed CSIT (see e.g. [6], [7], [21]). In contrast to these previous works, the
distributed nature of the CSIT is exploited here such that the interference terms are estimated
and transmitted from the TX having the most accurate CSIT, during the same channel use in
which the interference terms are generated.
IV. MAIN RESULTS
As one of the key observations made in this paper, we found that the DoF behavior in a
Network MIMO channel with distributed CSIT quite depends on the CSI quality regime. To this
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end, notions of “weak CSIT” regime and “arbitrary CSIT” regime are introduced to characterize
the interval in which the CSI scaling coefficients α(j) are allowed to take their values. We first
provide results for a given weak CSIT regime and then move on to the arbitrary CSIT regime.
We define a weak-CSIT regime as follows:
Definition 1. In the K-user Network MIMO with distributed CSIT and K ≥ 2, we define a weak
CSIT regime as comprising all the CSIT configurations satisfying that
αmax ≤ 1
1 +K(K − 2). (13)
In the two-user case, this condition reduces to α(j) ≤ 1,∀j ∈ {1, 2}, while in the three-user
case, CSIT is said to be weak if α(j) ≤ 1/4,∀j ∈ {1, 2, 3}, and so forth. Before detailing the
results of achievability in Subsections IV-B and IV-C, we start by providing an outerbound which
will be useful to interpret the following results.
Remark 5. As hinted in Section III, the weak CSIT regime it not uniquely defined. In fact, the
weak-CSIT regime considered above is larger than the one obtained using the simple scheme
described in Section III.
A. Centralized Outerbound
In the following, we prove rigorously the intuitive result that CSIT discrepancies between TXs
does not improve the DoF.
Theorem 1. In the K-user Network MIMO channel with distributed CSIT, the optimal DoF is
upperbounded by the DoF achieved in a centralized CSIT configuration in which all the TXs
can share perfectly their CSI estimates. Specifically, it holds that
DoFDCSI(α) ≤ DoFCCSI (αmax) (14)
where DoFCCSI(α) denotes the DoF achieved in a centralized CSIT configuration with the CSIT
scaling coefficient α [4].
Proof. This intuitive result is proved by considering the outerbound formed by a genie-aided
setting where all the channel estimates are perfectly shared between the TXs. This genie-aided
configuration is a centralized setting as all TXs share the same CSI. Furthermore, because of
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the assumption over the probability distribution of the channel in (5), it holds that
pH|H(1),...,H(K)(H) = O(√
Pαmax
). (15)
After having defined T , {H(1), . . . , H(K)} to represent the total available CSIT, it becomes
clear that it is possible to apply the outerbound derived in [4] for the centralized case. The DoF
of this genie-aided centralized setting is then equal to
DoFCCSI(αmax) = 1 + (K − 1)αmax. (16)
B. Weak CSIT Regime
Let us now consider the weak-CSIT regime defined in Definition 1.
Theorem 2. In the K-user Network MIMO with distributed CSIT, the optimal sum DoF in the
weak CSIT regime defined in Definition 1 satisfies
DoFDCSI(α) = 1 + (K − 1)αmax. (17)
Proof. The outerbound is obtained from Theorem 1 and a scheme achieving this outerbound is
described in Section VI.
In the weak CSIT regime defined, it is then sufficient in terms of DoF to provide a CSI estimate
at a single TX. This surprising result is in strong contrast with the performance obtained using
conventional ZF where the DoF is limited by the worst accuracy across all TXs (more precisely
the DoF is equal to 1 + (K− 1) minj α(j) when considering successive decoding). Note that this
is despite the fact that the ZF approach was recently shown to be DoF optimal for the BC under
centralized CSIT setting [4].
Remark 6. In the two-user case, the weak CSIT condition considered reduces to α(j) ≤ 1,∀j ∈
{1, 2}, such that the notion of weak CSIT coincides with the arbitrary regime. This is in
agreement with the result in [11] that it is possible to achieve 1 + (K − 1) max(α(1), α(2)
)in the 2-user case, for any value of α(1) and α(2).
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C. Achievable DoF for Arbitrary CSIT Regime with K = 3
Going beyond the weak-CSIT regime, we present below a new transmission scheme exploiting
the insights obtained in the weak CSIT regime to achieve a DoF well beyond the DoF achieved
using conventional transmission schemes. Deriving a transmission scheme for any CSIT config-
uration and any number of users is out of the scope of this work and is the topic of ongoing
research within our group.
Theorem 3. In the 3-user Network MIMO with distributed CSIT and α(1) ≥ α(2) ≥ α(3), it holds
that
DoFDCSI(α) ≥
1 + 2α(1) if α(1) ≤ 14
32α(1)−α(2)+2α(1)α(2)
4α(1)−α(2) if α(1) ≥ 14.
(18)
Proof. See the scheme description in Section VII.
Comparing this achievability result with the Centralized Outerbound in Theorem 1 gives the
following corollary.
Corollary 1. The lower bound provided in Theorem 3 for K = 3 users is tight in the weak CSIT
regime defined in Definition 1 and in the arbitrary CSIT regime if α(1) = α(2).
The DoF achieved with the proposed scheme is illustrated in Fig. 3. For α(1) ≤ 14, the
transmission occurs in the weak CSIT regime, and the proposed scheme coincides with the
transmission scheme for the weak CSIT, as described in Section VI. In that regime, the achieved
DoF only depends on the value of the best CSI scaling coefficient α(1), while for larger values
of α(1), the DoF also depends on the value of the second best CSI scaling coefficient α(2).
The proposed D-CSI robust transmission schemes rely on several ingredients which are (i)
Active-Passive (AP-) ZF precoding, (ii) interference quantization, and (iii) superposition coding.
AP-ZF was first introduced with a single so-called passive TX in [11] and we present below
a non-trivial generalization to an arbitrary number of passive TXs and an arbitrary number of
active TXs.
13
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
α(1)
Sum
DoF
Centralized OuterboundProposed schemeDoF using Conventional ZF [11]
α(2)=α(1) α(2)=α(1)/2
α(2)=0
Fig. 3: Sum DoF as a function of α(1). The DoF achieved is presented for exemplary values of
α(2), while the value of α(3) is set to 0 to emphasize the sensitivity with respect to one estimate.
V. PRELIMINARIES: ACTIVE-PASSIVE ZERO FORCING
Let us consider a setting in which K single-antenna TXs aim to transmit K−n data symbols
to one user (e.g., a user having K − n antennas) while zero-forcing interference to n other
single-antenna users, where 0 < n < K. Within this section, we denote the channel from the
K TXs to the interfered users by H ∈ Cn×K . We divide the TXs between so-called active TXs
and passive TXs and we consider without loss of generality that the first n TXs are the active
TXs while the remaining K − n TXs are the passive ones.
We define the active channel as the channel coefficients from the active TXs, denoted by
14
HA ∈ Cn×n, and the passive channel as the channel coefficients from the passive TXs, denoted
by HP ∈ Cn×(K−n), such that
H =[HA HP
]. (19)
This transmission setting considered to introduce AP-ZF is illustrated in Fig. 4.
n Active TXsK-n Passive
TXs
Kn x CH
n Interfered
users
Served user
n)-(Kn x P CHnn x A CH
K-n streams
Fig. 4: Illustration of the transmission scenario considered for AP-ZF.
Turning to the CSIT configuration, we assume that an estimate H(j) ∈ Cn×K is available
at TX j, for j ∈ [K]. We define the estimated active channel H(j)A ∈ Cn×n and the estimated
passive channel H(j)P ∈ Cn×(K−n) similarly to their perfectly known counterparts.
Let us now consider the signal processing at TX j. On the basis of the available CSIT H(j),
TX j computes TAPZF(j) ∈ CK×K−n where the part of the precoder which should be implemented
at the active TXs is denoted by λAPZFTA(j) ∈ Cn×K−n, where λAPZF is used to satisfy an average
sum power constraint (see its exact value in (23)), and is called the active precoder (computed
at TX j) while the part of the precoder which should be implemented at the passive TXs is
15
denoted by λAPZFTP ∈ C(K−n)×(K−n) and is called the passive precoder. It then holds that
TAPZF(j) = λAPZF
TA(j)
TP
. (20)
The precoder TP is arbitrarily chosen as any full rank matrix known to all TXs while the
precoder TA(j) is computed as
TA(j) =−((H
(j)A )HH
(j)A +
1
PIn
)−1(H
(j)A )HH
(j)P TP (21)
where P is the sum power of the data symbol transmitted. Note that the precoder TP is a CSI
independent precoder which is commonly agreed upon by all TXs beforehand.
The effective AP-ZF precoder is implemented in a distributed manner and is denoted by
TAPZF ∈ CK×K−n. It is a composite version of the precoders computed at each TX and is hence
given by
TAPZF , λAPZF
eH1 TA(1)
...
eHnTA(n)
TP
(22)
where ei ∈ Cn for i ∈ [n] is the ith row of the identity In and where the normalization
coefficient λAPZF is chosen as
λAPZF ,1√√√√√E
∥∥∥∥∥∥− (HH
AHA + 1P
In)−1
HHAHPTP
TP
∥∥∥∥∥∥2
F
. (23)
This normalization constant λAPZF requires only statistical CSI and can hence be applied at
every TX. It ensures that an average normalization constraint is satisfied, i.e., that
E[‖TAPZF‖2F
]= 1. (24)
Remark 7. The design of the Active precoder in (21) is an extension of the AP-ZF precoder
introduced in [11]. Intuitively, the active precoders invert the channel so as to cancel the
interference generated by the passive TXs. Note also that although TX j computes the full
precoder TAPZF(j), only some coefficients will be effectively used for the transmission due to
the distributed precoding configuration, as shown in (22).
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The key following properties can be easily shown from the precoder design.
Lemma 1. With perfect channel knowledge at all (active) TXs, the AP-ZF precoder with n active
TXs and K − n passive TXs satisfies
HTAPZF? −−−→P→∞
0n×(K−n) (25)
where TAPZF? denotes the AP-ZF precoder based on perfect CSIT and is given as
TAPZF? , λAPZF
TA?
TP
. (26)
Proof. Using the well known Resolvent identity [22, Lemma 6.1], we can write that(HH
AHA +1
PIn
)−1−(HH
AHA
)−1= −
(HH
AHA
)−1( 1
PIn
)(HH
AHA +1
PIn
)−1. (27)
We can then compute the leaked interference as
HTAPZF? = λAPZFHATA? + λAPZFHPTP (28)
(a)= λAPZFHA
(HH
AHA
)−1( 1
PIn
)(HH
AHA +1
PIn
)−1HH
AHPTP (29)
where equality (a) follows from inserting (27) inside the AP-ZF precoder and simplifying.
Letting the available power P tend to infinity, the leaked interference tends to zero.
Lemma 2. The AP-ZF precoder with n active TXs and K − n passive TXs is of rank K − n.
Proof. The passive precoder was chosen such that TP is full rank, i.e., of rank K − n. The
precoder TA(j) is a linear combination of TP for each j, such that the effective AP-ZF pre-
coder TAPZF resulting from distributed precoding is exactly of rank K − n.
Lemma 3. If H(j) , H+√P−α(j)∆(j) for α(j) ∈ [0, 1] with ∆(j) being drawn from a continuous
ergodic distribution with zero mean and bounded full rank covariance matrix, it then holds that∥∥HTAPZF∥∥2F≤ P−minj∈[n] α
(j)
. (30)
Proof. Following a similar approach as in [11], we can use once more the resolvent identity
[22, Lemma 6.1] to approximate the matrix inverse and show that∥∥TAPZF(j) −TAPZF?∥∥2F≤ P−α
(j)
. (31)
17
It then follows that ∥∥HTAPZF∥∥2F≤∥∥H (TAPZF −TAPZF?
)∥∥2F
(32)
≤ ‖H‖2F∥∥TAPZF −TAPZF?
∥∥2F
(33)
≤ ‖H‖2F
(n∑j=1
∥∥TAPZF(j) −TAPZF?∥∥2F
)(34)
≤ P−minj∈[n] α(j)
(35)
where (32) follows from Lemma 1 and (35) follows from Lemma 3.
The interpretation behind this result is that the interference attenuation of AP-ZF precoding is
only limited by the CSIT accuracy at the active TXs, and does not depend on the CSI accuracy
at the passive TXs.
Remark 8. Interestingly, it can be seen that the number of passive TXs determines the rank of
the precoder, i.e., the number of streams, and the number of active TXs determines the number
of ZF constraints that are satisfied.
VI. WEAK CSIT REGIME: ACHIEVABLE SCHEME
We now consider the weak CSIT regime and we describe the transmission scheme achieving
the DoF expression given in Theorem 2. Without loss of generality, we assume that the TX with
the best CSIT accuracy is TX 1, i.e., that α(1) = maxj∈[K] α(j).
A. Encoding
The proposed transmission scheme consists in only one channel use during which K− 1 data
symbols of rate α(1) log2(P ) bits are sent to each user (thus leading to K(K − 1) data symbols
being sent in one channel use), while an additional data symbol of data rate (1−α(1)) log2(P ) bits
is broadcast from TX 1. Note that the information which is contained in this broadcast symbol
is not only composed of information bits destined to one user, but is also composed of side
information necessary for decoding the other –private– data symbols, as will be detailed in
Subsection VI-B. The data symbol vector destined to user i is denoted by si ∈ CK−1 while the
broadcast data symbol is denoted by s0 ∈ C.
18
RX 1 RX 2 RX 3
3APZF3
H22
APZF2
H21
APZF1
H201,2 sThsThsTh sH3
APZF3
H12
APZF2
H11
APZF1
H101,1 sThsThsTh sH 3
APZF3
H32
APZF2
H31
APZF1
H301,3 sThsThsTh sH
)1(P
P P1-α(1)
Fig. 5: Illustration of the transmission scheme for the weak CSIT regime in the case of K = 3
users.
The transmitted signal x ∈ CK is then equal to
x =
1
0K−1×1
s0 +K∑i=1
TAPZFi si (36)
where
• si ∈ CK−1 contains K−1 data symbols of rate α(1) log2(P ) bits and power Pα(1)/(K(K−
1)) while TAPZFi ∈ CK×(K−1) is the AP-ZF precoder described in Section V, with TX 1
being the only active TX such that the interference are zero-forced at a single user, which
we choose to be user i+ 1 where i+ 1 = i mod [K] + 1.
• s0 ∈ C is a data symbol of rate (1 − α(1)) log2(P ) bits and power P − Pα(1) , and is
transmitted from TX 1 only.
The signal received at user i is then
yi=Hi,1s0︸ ︷︷ ︸.=P
+hHi TAPZF
i si︸ ︷︷ ︸.=Pα
(1)
+hHi
K∑k=1,k 6=i,k 6=i−1
TAPZFk sk︸ ︷︷ ︸
.=(K−2)Pα(1)
+hHi TAPZF
i−1 si−1︸ ︷︷ ︸.=P 0
(37)
where the noise term has been neglected for clarity. The last term in (37) scales as P 0 following
the attenuation by P−α(1) due to AP-ZF with TX 1 being the only active TX, as shown in
Lemma 3. The received signals during this transmission are illustrated in Fig. 5.
19
B. Interference Estimation and Quantization at TX 1
The data symbol s0 is used to convey to the users side information allowing to decode
their destined data symbols. More specifically, TX 1 uses its local CSIT H(1) to estimate the
interference terms hHi TAPZF
k sk for k 6= i, k 6= i− 1 that is going to be generated by the private
data symbols si,∀i ∈ [K]. Each interference term has a power scaling in Pα(1) such that using
α(1) log2(P ) bits, each term can be quantized with a quantization noise at the noise floor [20].
It can be seen by inspection that there are in total K(K − 2) such interference terms. In the
weak interference regime considered, it holds by definition that K(K − 2)α(1) ≤ 1− α(1) such
that these K(K− 2)α(1) log2(P ) bits can be transmitted via the common data symbol s0 of data
rate (1 − α(1)) log2(P ) bits. If the previous inequality is strict, these bits are completed with
information bits destined to any particular user.
C. Decoding and DoF Analysis
It remains to verify that this scheme leads to the claimed DoF. Let us consider without loss of
generality the decoding at user 1 as the decoding at the other users will follow with a circular
permutation of the user’s indices.
Using successive decoding, the data symbol s0 is decoded first, followed by s1. The data
symbol s0 of rate of (1−α(1)) log2(P ) bits can be decoded with a vanishing probability of error
as its SINR can be seen in (37) to scale as P 1−α(1) .
Upon decoding s0, the estimated interferences (h(1)1 )HTAPZF
k sk, k ∈ {2, . . . , K − 1} are
obtained (up to the quantization noise at the noise floor). It holds that
(h(1)1 )HTAPZF
k sk = hH1 TAPZF
k sk + P−α(1)
(δ(1)1 )HTAPZF
k sk︸ ︷︷ ︸.=P 0
(38)
such that subtracting the estimated interference from the received signals can be done perfectly
in terms of DoF.
Remark 9. Note that TX 1 knows perfectly the effective precoder used in the transmission as
he is the only active TX.
After having decoded s0 and subtracted the quantized interference terms, the remaining signal
at user 1 is then (up to the noise floor)
y1 = hH1 TAPZF
1 s1. (39)
20
The estimated interference terms (h(1)i )HT
APZF(1)1 s1, i = 3, . . . , K, which have been obtained
through s0, are then used by user 1 to form a virtual received vector yv1 ∈ CK−1 equal to
yv1 ,
hH
1
(h(1)3 )H
...
(h(1)K )H
TAPZF1 s1. (40)
Each component of yv1 has a SINR scaling in Pα(1) and the AP-ZF precoder is of rank K − 1
(See Lemma 2) such that user 1 can decode its desired K − 1 data symbols, each with the rate
of α(1) log2(P ) bits.
Considering all users, K(K − 1)α(1) log2(P ) bits are transmitted through the private data
symbols and (1−α(1)−K(K − 2)α(1)) log2(P ) bits to any particular user through the common
data symbol s0. Adding the two expressions yields the claimed DoF.
VII. ARBITRARY CSIT REGIME FOR K = 3
In the arbitrary CSIT setting, finding an efficient transmission scheme adapted to all the
possible CSIT configurations is made challenging by the fact that the CSIT configuration is
characterized by K CSIT scaling coefficients. Consequently, there are many different CSIT
regimes and the transmission scheme needs to be very adaptive. Therefore, we provide in the
following an efficient heuristic scheme for the case of K = 3 users, while the extension of the
same ideas to K users is left for further works.
A. Main Principle
The first phase of the scheme is exactly the same as the scheme presented for the weak CSIT
regime in Section VI. If the transmission occurs in the weak CSIT regime, then the scheme
coincides with the scheme presented above. However, outside the weak CSIT regime, it holds
that K(K−2)α(1) > 1−α(1) such that all it is not possible to transmit to each user sufficient side
information to decode its desired private data symbols. Consequently, we resort to a second phase
of the transmission, in successive channel uses, during which the interference terms computed
at TX 1 that could not yet be transmitted, are taken care of.
During the second phase, the private data symbols are transmitted using AP-ZF with 2 active
TXs (TX 1 and TX 2) and the power Pα(2) . The use of 2 Active TXs with a lower power has for
21
consequence that interferences generated remain at the noise floor. Indeed, AP-ZF with k active
TXs is able to ZF interference at k users (See Section V), and the interference attenuation is
limited by the worst accuracy across the Active TXs (See Lemma 3). As a consequence, this
second phase does not generate any additional interference (in terms of DoF), such that the
common data symbol can be used to retransmit solely the interference generated during the first
phase.
This second phase is repeated until sufficient side information (i.e., quantized estimated
interference terms) have been transmitted to the users to decode the private data symbols emitted
during the first phase.
B. Encoding
As explained above, only the transmission during the second phase needs to be described.
The transmitted signal is then given by
x =
1
0
0
s0 +3∑i=1
t′APZFi s′i (41)
where
• s′i ∈ C is a private data symbol destined to user i, with power Pα(2)/3 and rate α(2) log2(P ) bits.
The AP-ZF precoder t′APZFi is designed with 2 active TXs –TX 1 and TX 2– and with one
passive TX –TX 3–. Consequently, the interferences can be zero-forced at both interfered
users (See Section V).
• s′0 ∈ C is a common data symbol, with power P − Pα(2) and rate (1 − α(2)) log2(P ) bits,
transmitted from TX 1 only.
The received signals at the users are then given by
y′1 = H1,1s0︸ ︷︷ ︸.=P
+hH1 t′APZF1 s′1︸ ︷︷ ︸.=Pα
(2)
+hH1 t′APZF2 s′2 + hH
1 t′APZF3 s′3︸ ︷︷ ︸
.=P 0
y′2 = H2,1s0︸ ︷︷ ︸.=P
+hH2 t′APZF2 s′2︸ ︷︷ ︸.=Pα
(2)
+hH2 t′APZF1 s′1 + hH
2 t′APZF3 s′3︸ ︷︷ ︸
.=P 0
y′3 = H3,1s0︸ ︷︷ ︸.=P
+hH3 t′APZF3 s′3︸ ︷︷ ︸.=Pα
(2)
+hH3 t′APZF1 s′1 + hH
3 t′APZF2 s′2︸ ︷︷ ︸
.=P 0
(42)
22
RX 1 RX 2 RX 3
P
3APZF3
H12
APZF2
H11
APZF1
H101,1 ''''''' sthsthsth sH
)2(P
3APZF3
H22
APZF2
H21
APZF1
H201,2 ''''''' sthsthsth sH 3
APZF3
H32
APZF2
H31
APZF1
H301,3 ''''''' sthsthsth sH
P1-α(2)
Fig. 6: Illustration of the second phase of the transmission in the arbitrary CSIT regime for
K = 3.
where the noise realizations have been neglected. The last terms of the received signals scale as
P 0 due to the attenuation by P−α(2) from AP-ZF with TX 1 and TX 2 being Active TXs (See
Lemma 3).
This transmission scheme is illustrated in Fig. 6.
C. DoF Analysis
The common data symbol is decoded first and its contribution to the received signal is removed.
This is possible with a vanishing probability of error as the SNR at each user scales in P 1−α(2) .
Using successive decoding, each user can then decode with a vanishing probability of error its
desired data symbol from the received signal. Thus, a sum DoF equal to 3α(2) is achieved during
each channel use of the second phase.
This second phase lasts until the totality of the quantized estimated interferences have been
successfully broadcast, i.e., during d(4α(1) − 1)/(1 − α(2))e channel uses. The impact of the
ceiling operator is made arbitrary small by repeating the first phase n1 times and the second
phase n2 times, with n1 and n2 chosen such that (4α(1)− 1)/(1−α(2)) is arbitrarily close to its
next integer. Consequently, we omit in the following the ceiling operator for the sake of clarity.
A sum DoF of 6α(1) is achieved during the first phase (conditioned on the successful retrans-
mission of the quantized estimated interferences) while a sum DoF of 3α(2) is achieved during
each channel use of the second phase. As the second phase lasts for (4α(1) − 1)/(1 − α(2))
23
channel uses, the DoF achieved by the full transmission scheme is
DoF =6α(1) + 4α(1)−1
1−α(2) 3α(2)
1 + 4α(1)−11−α(2)
= 32α(1) − α(2) + 2α(1)α(2)
4α(1) − α(2)
(43)
which is the claimed DoF, and concludes the proof.
VIII. CONCLUSION
We have described a new D-CSIT robust transmission schemes improving over the DoF
achieved by conventional precoding approaches when faced with distributed CSIT. As a first step,
we have derived an outerbound for the DoF achieved with D-CSIT, coined as the Centralized
Outerbound, and consisting in a genie-aided setting where all the CSI versions are made available
at all TXs. We have then uncovered the surprising result that in a certain “weak CSIT regime”,
it is possible to achieve this Centralized Outerbound in a D-CSIT configuration with CSIT
handed at a single TX. The robust precoding schemes proposed rely on new methods such
as the estimation of the interference and their transmission from a single TX, and the AP-ZF
precoding with multiple Passive TXs and multiple Active TXs. These new methods have a strong
potential for improvement in other wireless configurations with distributed CSIT. Deriving an
optimal transmission scheme for an arbitrary number of users and an arbitrary CSIT configuration
is a challenging and interesting research problem.
IX. ACKNOWLEDGMENT
The authors are grateful to Petros Elia, Dirk Slock and Shlomo Shamai for interesting and
helpful discussions.
APPENDIX
For the sake of completeness, we start by recalling the following result on multivariate
Gaussian distribution.
Theorem 4. [23] Let X and Y be centered and jointly Gaussian with covariance matrix KXX
and KYY. Assume that KYY > 0. Then the conditional distribution of X conditional on Y = y
24
is a multivariate Gaussian of mean
E[XYH
]K−1YYy. (44)
and covariance matrix
KXX − E[XYH
]K−1YYE
[YXH
]. (45)
For ease of notation, we will in the following consider the vectorized versions of the channel
and channel estimates, which we denote by h and h(j),∀j ∈ [K], respectively. Applying
Theorem 4, the conditional distribution ph|h(1),...,h(K) is multivariate Gaussian. Our goal is to
compute the covariance matrix of this conditional distribution, denoted by K. Before writing it
explicitly, we introduce the shorthand notation I ∈ CK3×K2 as
I ,[IK2 . . . IK2
]T
︸ ︷︷ ︸Ktimes
(46)
and Σ ∈ CK3×K3 as
Σ ,
P−α
(1)IK2
. . .
P−α(K)
IK2
. (47)
With these notations, the covariance matrix K is then written using Theorem 4 as
K = IK2 − IT [I IT + Σ]−1
I (48)
(a)= IK2 − ITΣ−1I + ITΣ−1I
[IK2 + ITΣ−1I
]−1ITΣ−1I (49)
(b)= IK2 −
[IK2 + ITΣ−1I
]−1ITΣ−1I (50)
= IK2 −
[IK2 +
K∑j=1
Pα(j)
IK2
]−1( K∑j=1
Pα(j)
IK2
)(51)
=1
1 +∑K
j=1 Pα(j)
IK2 (52)
where equality (a) follows from the Matrix Inversion Lemma [24, Chapter 3.1.1] and equality (b)
follows from basic algebraic manipulations. Hence, the conditional probability density function
is Gaussian with the variance of its elements scaling in P−αmax such that it satisfies that
max(ph|h(1),...,h(K)(h)).=√Pαmax . (53)
25
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